Hexadecimal as Shorthand for Binary

The number 218 expressed in binary is 11011010B. Expressed in hex, however, the same value is quite compact: DAH. The two hex digits comprising DAH merit a closer look. AH (or 0AH as your assembler will require it for reasons I explain later) represents 10 decimal. Converting any number to binary simply involves detecting the powers of two within it. The largest power of 2 within 10 decimal is 8. Jot down a 1 digit and subtract 8 from 10. What's left is 2. Now, 4 is a power of 2, but there is no 4 hiding within 2, so we put a 0 to the right of the 1. The next smallest power of 2 is 2, and there is a 2 in 2. Jot down another 1 to the right of the 0. Two from 2 is 0, so there are no 1s left in the number. Jot down a final 0 to the right of the rest to represent the 1s column. What you have is this:

1 0 1 0

Look back at the binary equivalent of 218: 11011010. The last four digits are 1010—the binary equivalent of 0AH.

The same will work for the upper half of DAH. If you work out the binary equivalence for 0DH as we just did (and it would be good mental exercise), it is 1101. Look at the binary equivalent of 218 this way:

218 decimal

1101 1010 binary

D A hex

It should be dawning on you that you can convert long strings of binary 1s and 0s into more compact hex format by converting every four binary digits (starting from the right, not from the left!) into a single hex digit.

As an example, here is a 32-bit binary number that is not the least bit remarkable:

11110000000000001111101001101110

This is a pretty obnoxious collection of bits to remember or manipulate, so let's split it up into groups of four from the right:

1111 0000 0000 0000 1111 1010 0110 1110

Each of these groups of four binary digits can be represented by a single hexadecimal digit. Do the conversion now. What you should get is the following:

1111 0000 0000 0000 1111 1010 0110 1110

F

0

0

0

F

A

6

E

In other words, the hex equivalent of that mouthful is

F000FA6E

In use, of course, you would append the H on the end, and also put a 0 at the beginning, so in any kind of assembly language work the number would actually be written 0F000FA6EH.

This is still a good-sized number, but unless you're doing things like counting hard drive space or other high-value things, such 32-bit numbers are the largest quantities you would typically encounter in journeyman-level assembly language programming.

Suddenly, this business starts looking a little more graspable.

Hexadecimal is the programmer's shorthand for the computer's binary numbers.

This is why I said earlier that computers use base 2 (binary) and base 16 (hexadecimal) both at the same time in a rather schizoid fashion. What I didn't say is that the computer isn't really the schizoid

one; you are. At their very hearts (as I explain in Chapter 3) computers use only binary. Hex is a means by which you and I make dealing with the computer easier. Fortunately, every four binary digits may be represented by a hex digit, so the correspondence is clean and comprehensible.

Prepare to Compute

Everything up to this point has been necessary groundwork. I've explained conceptually what computers do and have given you the tools to understand the slightly alien numbers they use. But I've said nothing so far about what computers actually are, and it's well past time. We return to hexadecimal numbers again and again in this book; I've said nothing thus far about hex multiplication or bit-banging. The reason is plain: Before you can bang a bit, you must know where the bits live. So, let's lift the hood and see if we can catch a few in action.

Chapter 3: Lifting the Hood Discovering What

Computers Actually Are

RAXie, We Hardly Knew Ye...

In January 1970 I was on the downwind leg of my senior year in high school, and the Chicago Public Schools had installed a computer somewhere. A truckful of these fancy typewriter gimcracks was delivered to Lane Tech, and a bewildered math teacher was drafted into teaching computer science (they had the nerve to call it) to a high school full of rowdy males.

I figured it out fairly quickly. You pounded out a deck of these goofy computer cards on the card punch machine, dropped them into the hopper of one of the typewriter gimcracks, and watched in awe as the typewriter danced its little golfball over the greenbar paper, printing out your inevitable list of error messages. It was fun. I got straight A's. I even kept the first program I ever wrote that did something useful: a little deck of cards that generated a table of parabolic correction factors for hand-figuring telescope mirrors, astronomy being my passion at the time. (The card deck is still in its place of honor on the narrow shelf here in my second-floor office, next to my 8-inch reel-to-reel tape deck and my father's venerable slide rule.)

The question that kept gnawing at me was exactly what sort of beast RAX (the computer's wonderfully appropriate name) actually was. What we had were ram-charged typewriters that RAX controlled over phone lines-that much I understood. But what was RAX itself?

I asked the instructor. In brief, the conversation went something like this:

ME: "Umm, sir, what exactly is RAX?"

HE: "Eh? Um, a computer. An electronic computer."

ME: "That's what it says on the course notes. But I want to know what RAX is made of and how it works."

HE: "Well, I'm sure RAX is all solid-state."

ME: "You mean, there's no levers and gears inside."

HE: "Oh, there may be a few. But no vacuum tubes."

ME: "I wasn't worried about tubes. I suppose it has a calculator in it somewhere. But what makes it remember that A comes before B? How does it know what FORMAT means? How does it tell time? What does it have to do to answer the phone?"

HE: "Now, come on, that's why computers are so great! They put it all together so that we don't have to worry about that sort of thing! Who cares what RAX is? RAX knows FORTRAN and will execute any correct FORTRAN program. That's what matters, isn't it?"

He was starting to sweat. So was I. End of conversation.

That June I graduated with three inches of debugged and working FORTRAN punch cards in my bookbag, and still had absolutely no clue as to what RAX was.

It has bothered me to this day.

Gus to the Rescue

I was thinking about RAX six years later, while on the Devon Avenue bus heading for work, with the latest copy of Popular Electronics in my lap. The lead story involved a little thing called the COSMAC ELF, which consisted of a piece of perfboard full of integrated circuit chips, all wired together, plus some toggle switches and a pair of LED numeric displays.

It was a computer. (Said so right on the label, heh.) The article told us how to put it together, and that was about all. What did those chips do? What did the whole thing do? It was driving me nuts.

As usual, my friend Gus Flassig got on the bus at Ashland Avenue and sat down beside me. I asked him what the damned thing did. He was the first human being to make the concept hang together for me:

"These are memory chips. You load numbers into the memory chips by flipping these switches in different code patterns. Each number means something to the CPU chip. One number makes it add; another number makes it subtract; another makes it write different numbers into memory, and lots of other things. A program consists of a bunch of these instruction-numbers in a row in memory. The computer reads the first number, does what the number tells it to do, and then reads the second one, does what that number says to do, and so on until it runs out of numbers."

If you don't find that utterly clear; don't worry. I had had the advantage of being an electronics hobbyist (so I knew what some of the chips did) and had already written some programs in RAX's FORTRAN. But for me, my God, everything suddenly hit critical mass and exploded in my head until the steam started pouring out of my ears.

No matter what RAX was, I knew that it had to be something like the COSMAC ELF on a larger scale. I built an ELF. It was quite an education, and allowed me to understand the nature of computers at a very deep level. I don't recommend that anybody but total crazies wirewrap their own machines out of loose chips anymore, although it was a common enough thing to do in the mid-late seventies.

As a sidenote, someone has written a Windows-based simulation of the COSMAC ELF that looks just like the one I built, and will actually accept and execute COSMAC programs. It's a lot of fun and might give you some perspective on what passed for computing in early 1976. The URL is as follows:

www.incolor.inetnebr.com/bill_r/computer_simulators.htm

The site's author, Bill Richman, has also reprinted the Popular Electronics article that I built the device from. All fascinating reading-and a very good education in the deepest silicon concepts underlying computing as it was then and remains to this day.

In this chapter I try and provide you with some of the insights that I obtained while assembling my own machine the hard way. (You wonder where the "hard" in "hardware" comes from? Not from the sound it makes when you bang it on the table, promise...)